From what I hear from other teachers, there isn't a huge amount of teaching work available at the moment in Rome. To get well-paid work, you generally need to have experience and qualifications, as well as a permesso di soggiorno. If you don't have EU citizenship or a working visa, it is going to be difficult to get work in Rome at a decent school.

You could get some private work and be paid cash in hand, but if you are after an ESL career, I would suggest trying to get experience in a country where it's easy for Americans to get a working visa.